Scalar Field Variables Research Articles

Thermal field reconstruction and compressive sensing using proper orthogonal decomposition

Model order reduction allows critical information about sensor placement and experiment design to be distilled from raw fluid mechanics simulation data. In many cases, sensed information in conjunction with reduced order models can also be used to regenerate full field variables. In this paper, a proper orthogonal decomposition (POD) inferencing method is extended to the modeling and compressive sensing of temperature, a scalar field variable. The method is applied to a simulated, critically stable, incompressible flow over a heated cylinder (Re = 1000) with Prandtl number varying between 0.001 and 50. The model is trained on pressure and temperature data from simulations. Field reconstructions are then generated using data from selected sensors and the POD model. Finally, the reconstruction error is evaluated across all Prandtl numbers for different numbers of retained modes and sensors. The predicted trend of increasing reconstruction accuracy with decreasing Prandtl number is confirmed and a Prandtl number/sensor count error matrix is presented.

Emergence of precursor instabilities in geo-processes: Insights from dense active matter

We present the hypothesis that investigation of precursor mechanisms to large scale instabilities, that have so far been overlooked in geo-processes, is possible. These precursor processes are evident in multicomponent materials, such as granular matter, when driven far from equilibrium on its microscale. The material is then classified as “dense active matter” with unexpected behaviour by non-local dissipation of internal energy releasing its dynamic incompatibility with the macroscopic gradients as self-excitation waves under external forcing. These instabilities are known in solid mechanics as flutter instabilities, nucleating at what is more widely known as an “exceptional point” in a variety of systems when two or more eigenvalues of the system coalesce. The common principle to connect processes at and across their characteristic scales is investigated using a minimalist formulation by coupling the scalar field variables of solid and fluid pressures in a compacting porous medium. We present a multiphysics generalisation of the phenomenon to the exciting findings of fluctuations with oscillatory exponential growth which nucleate at the exceptional point for inception of complex conjugate eigenmodes and propose a rigorous theory based on the extension of Onsager's theorem to non-local processes. Future work will need to compare model predictions to carefully designed laboratory experiments and expand the work to bridge the scale of the laboratory to the scale of field applications including design of new sensors tuned for detecting exceptional points preceding collapse of materials.

Quantum Fractionary Cosmology: K-Essence Theory

Using a particular form of the quantum K-essence scalar field, we show that in the quantum formalism, a fractional differential equation in the scalar field variable, for some epochs in the Friedmann–Lemaı^tre–Robertson–Walker (FLRW) model (radiation and inflation-like epochs, for example), appears naturally. In the classical analysis, the kinetic energy of scalar fields can falsify the standard matter in the sense that we obtain the time behavior for the scale factor in all scenarios of our Universe by using the Hamiltonian formalism, where the results are analogous to those obtained by an algebraic procedure in the Einstein field equations with standard matter. In the case of the quantum Wheeler–DeWitt (WDW) equation for the scalar field ϕ, a fractional differential equation of order β=2α2α−1 is obtained. This fractional equation belongs to different intervals, depending on the value of the barotropic parameter; that is to say, when ωX∈[0,1], the order belongs to the interval 1≤β≤2, and when ωX∈[−1,0), the order belongs to the interval 0<β≤1. The corresponding quantum solutions are also given.

Mixing and combustion in a laminar shear layer with imposed counterflow

Abstract

Combustion with Multiple Flames under High Strain Rates

ABSTRACT Structures with multiple flames, nonpremixed and premixed, in strained flow representative of turbulent combustion are analyzed with reacting, viscous, three-dimensional counterflows. There can be differing normal strain rates in each of the three directions. Special attention is given to configurations with a diffusion flame plus one or two partially premixed flames. Reduction of the equations to a similar form is obtained allowing for variations in density due to temperature, heat conduction, and mass diffusion. Solutions to the Navier-Stokes equations are obtained without the boundary-layer approximation. In steady, variable-density configurations, a set of ODEs governs the two transverse velocity components and the axial velocity component, as well as the scalar-field variables. Results for the velocity and scalar fields are found for a full range of the distribution of normal strain rates between the two transverse directions, a very wide range of Damköhler number Da based on strain rate, and various Prandtl number values. As strain rate is increased and/or pressure decreases, the multiple flames will merge into a single diffusion flame. With further changes in that direction, extinction will occur. In the three-flame case, the fuel-rich premixed flame merges first with the diffusion flame as Da decreases; then with further decrease, the fuel-lean premixed flame merges with the diffusion flame, leaving just a diffusion flame that will extinguish with further decrease. Velocity overshoots are seen in the viscous layer, yielding an important correction of theories based on a constant-density assumption. Advection and convection are important in the neighborhood of the flames.

Counterflow and wall stagnation flow with three-dimensional strain

Three-dimensional (3D) viscous counterflows and wall stagnation flows are analyzed with differing normal strain rates in each of the three directions. Reduction of the equations to a similar form is obtained allowing for variations in density due to temperature and composition, heat conduction, and, for the counterflow, mass diffusion and the presence of a flame. Solutions to the Navier-Stokes equations are obtained without the boundary-layer approximation. For the steady and unsteady incompressible counterflows, analytical solutions are obtained for the flow field and the scalar fields subject to heat and mass transfer. In steady, variable-density configurations, a set of ordinary differential equations (ODEs) governs the two transverse velocity and the axial velocity profiles as well as the scalar-field variables. Diffusion rates for mass, momentum, and energy depend on the two normal strain rates parallel to the counterflow interface or the wall and thereby not merely on the sum of those two strain rates. For thin diffusion flames, the location, burning rate, and peak temperature are readily obtained. Solutions for planar flows and axisymmetric flows are obtained as limits here. Results for the velocity and scalar fields are found for a full range of the distribution of normal strain rates between the two transverse directions, various Prandtl number values, and various ambient (or wall) temperatures. For counterflows with flames and stagnation layers with hot walls, velocity overshoots are seen in the viscous layer, yielding an important correction of theories based on a constant-density assumption.

Computational investigation of a lifted hydrogen flame with LES and FGM

A numerical analysis of an atmospheric, subsonic, turbulent lifted H2/N2 jet flame in vitiated co-flow is presented. Turbulence is treated by a Large Eddy Simulation (LES) methodology. As the turbulent combustion model, the Flamelet Generated Manifold (FGM) approach is used, which enables the incorporation of detailed chemistry via two additional scalar field variables. The results are compared with the measurements and with the predictions of other authors. It is observed that the achieved predictive capability is, in general, quite fair and comparable to that of alternative turbulent combustion models. This demonstrates the predictive capability of the FGM for this class of problems, and shows that the method, which is comparably very cost-effective in incorporating detailed combustion chemistry in turbulent flame calculations, can be applied, in combination with LES, to predict lifted hydrogen flames.

Superconvergence in a DPG method for an ultra-weak formulation

In this work we study a DPG method for an ultra-weak variational formulation of a reaction–diffusion problem. We improve existing a priori convergence results by sharpening an approximation result for the numerical flux. By duality arguments we show that higher convergence rates for the scalar field variable are obtained if the polynomial order of the corresponding approximation space is increased by one. Furthermore, we introduce a simple elementwise postprocessing of the solution and prove superconvergence. Numerical experiments indicate that the obtained results are valid beyond the underlying model problem.

An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation

We present an explicit hybridizable discontinuous Galerkin (HDG) method for numerically solving the acoustic wave equation. The method is fully explicit, high-order accurate in both space and time, and coincides with the classic discontinuous Galerkin (DG) method with upwinding fluxes for a particular choice of its stabilization function. This means that it has the same computational complexity as other explicit DG methods. However, just as its implicit version, it provides optimal convergence of order k+1 for all the approximate variables including the gradient of the solution, and, when the time-stepping method is of order k+2, it displays a superconvergence property which allow us, by means of local postprocessing, to obtain new improved approximations of the scalar field variables at any time levels for which an enhanced accuracy is required. In particular, the new approximations converge with order k+2 in the L2-norm for k≥1. These properties do not hold for all numerical fluxes. Indeed, our results show that, when the HDG numerical flux is replaced by the Lax–Friedrichs flux, the above-mentioned superconvergence properties are lost, although some are recovered when the Lax–Friedrichs flux is used only in the interior of the domain. Finally, we extend the explicit HDG method to treat the wave equation with perfectly matched layers. We provide numerical examples to demonstrate the performance of the proposed method.

Three dimensional thermal-solute phase field simulation of binary alloy solidification

We employ adaptive mesh refinement, implicit time stepping, a nonlinear multigrid solver and parallel computation to solve a multi-scale, time dependent, three dimensional, nonlinear set of coupled partial differential equations for three scalar field variables. The mathematical model represents the non-isothermal solidification of a metal alloy into a melt substantially cooled below its freezing point at the microscale. Underlying physical molecular forces are captured at this scale by a specification of the energy field. The time rate of change of the temperature, alloy concentration and an order parameter to govern the state of the material (liquid or solid) are controlled by the diffusion parameters and variational derivatives of the energy functional. The physical problem is important to material scientists for the development of solid metal alloys and, hitherto, this fully coupled thermal problem has not been simulated in three dimensions, due to its computationally demanding nature. By bringing together state of the art numerical techniques this problem is now shown here to be tractable at appropriate resolution with relatively moderate computational resources.

Conceptual Difficulties in Plasticity including the Gradient of one Scalar Plastic Field Variable

AbstractThis article discusses a conceptual problem of gradient plasticity theories including a scalar plastic field variable, the Laplacian of which enters the yield criterion. Since the plastic field variable is assumed a scalar, it contains no information on the direction of the plastic flow. The work at hand demonstrates that even arbitrarily small perturbations can determine the direction of the plastic deformation in many formulations, which is considered an unphysical behavior. In this sense, the solution is not stable with respect to the boundary conditions. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Exploring scalar field dynamics with Gaussian processes

The origin of the accelerated expansion of the Universe remains an unsolved mystery inCosmology. In this work we consider a spatially flat Friedmann-Robertson-Walker (FRW) Universe withnon-relativistic matter and a single scalar field contributing to the energy density of the Universe.Properties of this scalar field, like potential, kinetic energy, equation of state etc. arereconstructed from Supernovae and BAO data using Gaussian processes. We also reconstruct energy conditions and kinematicvariables of expansion, such as the jerk and the slow roll parameter. We find that the reconstructed scalar field variables and the kinematic quantities are consistent with a flat ΛCDM Universe. Further, we find that the null energy condition is satisfied for the redshift range of the Supernovae data considered in the paper, but the strong energy condition is violated.

A Phase Field Approach for Dynamic Fracture

AbstractIn phase field fracture models cracks are indicated by the value of a scalar field variable which interpolates smoothly between broken and undamaged material. The evolution equation for this crack field is coupled to the mechanical field equations in order to model the mutual interaction between the crack evolution and mechanical quantities. In finite element simulations of crack growth at comparatively slow loading velocities, a quasi‐static phase field model yields reasonable results. However, the simulation of fast loading or the nucleation of new cracks challenges the limits of such a formulation. Here, the quasi‐static phase field model predicts brutal crack extension with an artificially high crack speed. In this work, we analyze to which extend a dynamic formulation of the mechanical part of the phase field model can overcome this paradox created by the quasi‐static formulation. In finite element simulations, the impact of the dynamic effects is studied, and differences between the crack propagation behavior of the quasi‐static model and the dynamic formulation are highlighted. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Background independent quantizations—the scalar field: II

We are concerned with the issue of the quantization of a scalar field in a diffeomorphism invariant manner. We apply the method used in loop quantum gravity. It relies on the specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the ‘quantum’ polymer ∗-star algebra and looking for positive linear functionals, called states. As assumed in our paper, homeomorphism invariance allows us to derive the complete class of the states. They are determined by the homeomorphism invariant states defined on the CW-complex ∗-algebra. The corresponding GNS representations of the polymer ∗-algebra and their self-adjoint extensions are derived, the equivalence classes are found, and invariant subspaces characterized. In part I we outlined those results. Here, we present the technical details.

Background independent quantizations—the scalar field: I

We are concerned with the issue of quantization of a scalar field in a diffeomorphism-invariant manner. We apply the method used in loop quantum gravity. It relies on a specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the ‘quantum’ polymer *-star algebra and looking for positive linear functionals, called states. It is assumed in this paper that homeomorphism invariance allows us to determine a complete class of the states. Except one, all of them are new. In this paper we outline the main steps and conclusions, and present the results: the GNS representations, characterization of those states which lead to essentially self-adjoint momentum operators (unbounded), identification of the equivalence classes of the representations as well as of the irreducible ones.

Reduced models for fluid–structure interaction problems

AbstractSeveral so‐called reduced order formulations are reviewed for linear vibration analysis of bounded fluid–structure systems for low modal density situations. Compressibility effects in the fluid for interior structural–acoustic problems, and free surface gravity effects, for hydroelastic–sloshing interaction problems in the case of incompressible liquids, are examined. Reduced order models leading to symmetric matrix systems are then described using the static well‐posed behaviour of the irrotational fluid. In this respect, the fluid–structure boundary value local equations, expressed in terms of fluid scalar field variables for the fluid (and displacement variables for the structure) are regularized for zero‐frequency limit. Copyright 2004 John Wiley & Sons, Ltd.

Prediction of Lifted Diffusion Flames by using the Scalar Field Variable.

By incorporating the scalar field variable, G, into the numerical method with the flame sheet model, one of reaction models of simulation of diffusion flames, we have developed a new calculation method of lifted flames. The present method can deal with the flame instabilities including the blowoff, liftoff, and extinction, while the conventional flame sheet model cannot deal with them, because scalar quantities are decided uniquely against the mixture fraction in that model. The burning velocity included in the production term of the equation of scalar field variable, G, is formulated as a function of the scalar dissipation rate. This makes the treatment of the flame instabilities possible. The frozen scalar structure is applied to the upstream of the intersection of isolines of the stoichiometric line and G=1, the transitional structure expressed by a weight function based on the scalar field variable, G, to a small zone behind the intersection, and the diffusion flame structure to the downstream region. Numerical results of a reactive two dimensional mixing layer by the present method are in reasonable agreement with results of direct numerical simulations with a finite reaction rate and irreversible one step reaction model.

Coupled map lattices simulating hydrodynamical turbulence

Cascade models driven by strongly chaotic driving forces yield a successful description of velocity probability densities and scaling exponents measured in fully developed three-dimensional turbulence. We prove that for these types of models velocity differences are bounded, in agreement with measurements in turbulence experiments. Regarding the radial component of the velocity difference as a strongly fluctuating scalar field variable, we show how a dynamics governed by coupled quadradic or cubic mappings can arise from a self-interacting momentum balance equation. The spatial coupling of the coupled map lattice at the top of the cascade is a measure for the inverse Reynolds number. Some numerical results on force patterns are presented.

Cauchy-characteristic evolution of Einstein-Klein-Gordon systems.

A Cauchy-characteristic initial value problem for the Einstein-Klein-Gordon system with spherical symmetry is presented. Initial data are specified on the union of a space-like and null hypersurface. The development of the data is obtained with the combination of a constrained Cauchy evolution in the interior domain and a characteristic evolution in the exterior, asymptotically flat region. The matching interface between the space-like and characteristic foliations is constructed by imposing continuity conditions on metric, extrinsic curvature and scalar field variables, ensuring smoothness across the matching surface. The accuracy of the method is established for all ranges of $M/R$, most notably, with a detailed comparison of invariant observables against reference solutions obtained with a calibrated, global, null algorithm.

Quantum gravitational collapse of a scalar field and the wave function of black hole decay.

We develop a quantum description of spherically symmetric gravitational collapse of a massless scalar field. The canonical quantization procedure is applied to a minisuperspace model in which the four-metric and scalar field variables are restricted to a self-similar functional form. Our main purpose is to propose a mechanism of black hole decay in terms of a time evolution of the wave function defined on the minisuperspace. For the self-similar classical dynamics there exists a one-parameter family of solutions for the Einstein-scalar equations, and the Hamiltonian turns out to play the role of the critical parameter which separates the supercritical solution corresponding to a black hole formation from the subcritical one corresponding to a wave reflection to infinity. We derive a set of the eigenfunctions of the quantum Hamiltonian operator, which is interpreted to be the quantum version of the supercritical and subcritical solutions. One of the metric components is also treated as a quantum operator which does not commute with the Hamiltonian operator. The expansion of the eigenfunction of the metric operator in terms of the Hamiltonian eigenfunctions motivates us to introduce the superposition principle and the time-dependent Schr\odinger equation which lead to the uncertainty of the Hamiltonian eigenvalue. Then we study the time evolution of the initial black hole state, which is chosen to be one of the supercritical Hamiltonian eigenstates. We arrive at the conclusion that the time evolution toward black hole decay is essentially due to the breakdown of orthogonality of the Hamiltonian eigenfunctions: The quantum black hole behaves like a wave packet written by a superposition of the various subcritical states. The localized structure of the black hole wave packet begins to spread as each subcritical state evolves with time according to the Schr\odinger equation. We can estimate the decay time and find the final behavior of the wave function describing an outgoing flux of scalar waves observable at future null infinity.